Suppose that we have a Euclidean Domain as a ring, but we are not given a Euclidean function. We know that the ring is a Euclidean Domain. We have an oracle for +, . and whether a/b. Can we construct a computable function v, that will satisfy the definision of a Euclidean function?
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This question and its answer seem to answer your question in negative. – Peter Košinár Jun 07 '13 at 16:51